The shallow water model is one of the important models in dynamical systems. This paper investigates the adaptive chaos control and synchronization of the shallow water model. First, adaptive control laws are designed to stabilize the shallow water model. Then adaptive control laws are derived to chaos synchronization of the shallow water model. The sufficient conditions for the adaptive control and synchronization have been analyzed theoretically, and the results are proved using a Barbalat's Lemma.

A dynamical system is a system that changes over time. Chaotic systems are dynamical systems that are highly sensitive to initial conditions. Chaos phenomena in weather models were first observed by Lorenz equation; a large number of chaos phenomena and chaos behavior have been discovered in physical, social, economical, biological, and electrical systems.

Atmosphere is a dynamical system. An atmospheric model is a set of equations that describes behavior of the atmosphere. The shallow water model is simple model for the atmosphere. Shallow water model is the set of the equations of motion that describes the evolution of a horizontal structure, hydrostatic homogeneous, and incompressible flow on the sphere [

The control of chaotic systems is to design state feedback control laws that stabilize the chaotic systems. Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The usual objective of control theory is to calculate solutions for the proper corrective action from the controller that result in system stability.

Synchronization of chaotic systems is phenomena that may occur when two or more chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator, because of the butterfly effect, which causes the exponential divergence of the trajectories of two identical chaotic systems started with nearby the same initial conditions. Synchronizing two chaotic systems is seemingly a very challenging problem in chaos literature [

In 1990, Pecora and Caroll [

In most of the chaos synchronization approaches, the drive-response formalism has been used. If a particular chaotic system is called the drive system and another chaotic system is called the response system, then the idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system tracks the output of drive system asymptotically stable.

This paper is organized as follows. Section

Consider a nonlinear nonautonomous differential equation of the general form

The zero solution of (

The zero solution of (

Consider the control system

The control system (

Consider two nonlinear systems

Response system and drive system are said to be

For nonautonomous system,

A chaotic system has complex dynamical behaviors; those posses some special features, such as being extremely sensitive to tiny variations of initial conditions. In this section, adaptive control method is applied to control chaos shallow water model.

Shallow water model is the set of the equations of motion that describes the evolution of a horizontal structure, hydrostatic homogeneous, and incompressible flow on the sphere. Euler’s equations of motion of an ideal fluid are as follows:

Consider the controlled system of (

In this case the control law is

For

Let us consider the Lyapunov function

In this section, the adaptive synchronization is introduced to make two of the shallow water model. The sufficient condition for the synchronization has been analyzed theoretically, and the result is proved using a Barbalat’s Lemma. Assume that there are two shallow water models such that the drive system is to control the response system. The drive and response system are given as

Let

It is easy to see from (

In this paper, we applied adaptive control theory for the chaos control and synchronization of the shallow water model. First, we designed adaptive control laws to stabilize the shallow water model based on the adaptive control theory and stability theory. Then, we derived adaptive synchronization to the shallow water model. The sufficient conditions for the adaptive control and synchronization of the shallow water model have been analyzed theoretically, and the results are proved using a Barbalat’s Lemma.